Let Sn be the director circle of the circle Sn-1- where n- 2- S1-x2-y2-1- The equation of the circle orthogonal to the circle S10 is given by x2-y2-2gx-2fy-c-0- Then

S_1:x^2+y^2=1 S_2:x^2+y^2=2 S_3:x^2+y^2=2^2 ∴ nbsp;S_n:x^2+y^2=2^n 1 ⇒ nbsp;S_10:x^2+y^2=2^9 x^2+y^2 +2g_1x +2f_1y+c_1=0 x^2+y^2 +2g_2x +2f_2y+c_2=0 Cond ...

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Byju's Answer

Standard XII

Mathematics

Equation of Tangent in Slope Form

Let Sn be the...

Question

Let $S_{n}$ be the director circle of the circle $S_{n - 1}$ , where $n \geq 2$ . $S_{1} : x^{2} + y^{2} = 1$ . The equation of the circle orthogonal to the circle $S_{10}$ is given by $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . Then

the value of

c

cannot be determined

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no such circle is possible

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there is a unique value of

c

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the value of

c

depends upon

g

and

f

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Solution

The correct option is C there is a unique value of $c$
$S_{1} : x^{2} + y^{2} = 1 S_{2} : x^{2} + y^{2} = 2 S_{3} : x^{2} + y^{2} = 2^{2} ∴ S_{n} : x^{2} + y^{2} = 2^{n - 1} \Rightarrow S_{10} : x^{2} + y^{2} = 2^{9}$

$x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0$
$x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$
Condition of orthogonal is,
$2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

$∴ 2 (0 + 0) = c - 2^{9}$
$\Rightarrow c = 512$

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Similar questions

Q. Let

S_{n}

be the director circle of the circle

S_{n - 1}

, where

n \geq 2

S_{1} : x^{2} + y^{2} = 1

. The equation of the circle orthogonal to the circle

S_{10}

is given by

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

. Then

Q. The condition if the equation of a circle concentric with the circle

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

x^{2} + y^{2} + 2 g x + 2 f y + k = 0

, where

k

is a constant.

Which of the following equations represents a circle with centre $(g, f)$ and radius $\sqrt{g^{2} + f^{2} + c}$ ?

Q. Length of tangent drawn from any point of circle

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

to the circle

x^{2} + y^{2} + 2 g x + 2 f y + d = 0

, (d > c) is

Q. Length of the tangent drawn from any point on the circle

x^{2} + y^{2} + 2 g x + 2 f y + c_{1} = 0

to the circle is

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

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Let Sn be the director circle of the circle Sn−1, where n≥2. S1:x2+y2=1. The equation of the circle orthogonal to the circle S10 is given by x2+y2+2gx+2fy+c=0. Then

The correct option is C there is a unique value of cS1:x2+y2=1S2:x2+y2=2S3:x2+y2=22∴Sn:x2+y2=2n−1⇒S10:x2+y2=29 x2+y2+2g1x+2f1y+c1=0 x2+y2+2g2x+2f2y+c2=0 Condition of orthogonal is, 2g1g2+2f1f2=c1+c2 ∴2(0+0)=c−29 ⇒c=512

Let $S_{n}$ be the director circle of the circle $S_{n - 1}$ , where $n \geq 2$ . $S_{1} : x^{2} + y^{2} = 1$ . The equation of the circle orthogonal to the circle $S_{10}$ is given by $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . Then